Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:
(i) \(4x^2 - 3x + 7\)
(ii) \(y^2 + \sqrt{2}\)
(iii) \(3\sqrt{t} + t\sqrt{2}\)
(iv) \(y + \dfrac{2}{y}\)
(v) \(x^{10} + y^3 + t^{50}\)
(i) and (ii) are polynomials in one variable.
(v) is a polynomial in three variables.
(iii) and (iv) are not polynomials because their variable exponents are not whole numbers.
Write the coefficients of \(x^2\) in each of the following:
(i) \(2 + x^2 + x\)
(ii) \(2 - x^2 + x^3\)
(iii) \(\dfrac{\pi}{2} x^2 + x\)
(iv) \(\sqrt{2}x - 1\)
(i) 1
(ii) -1
(iii) \(\dfrac{\pi}{2}\)
(iv) 0
Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Example binomial of degree 35: \(3x^{35} - 4\)
Example monomial of degree 100: \(\sqrt{2} \, y^{100}\)
Write the degree of each of the following polynomials:
(i) \(5x^3 + 4x^2 + 7x\)
(ii) \(4 - y^2\)
(iii) \(5t - \sqrt{7}\)
(iv) \(3\)
(i) 3
(ii) 2
(iii) 1
(iv) 0
Classify the following as linear, quadratic, and cubic polynomials:
(i) \(x^2 + x\)
(ii) \(x - x^3\)
(iii) \(y + y^2 + 4\)
(iv) \(1 + x\)
(v) \(3t\)
(vi) \(r^2\)
(vii) \(7x^3\)
(i) quadratic
(ii) cubic
(iii) quadratic
(iv) linear
(v) linear
(vi) quadratic
(vii) cubic
Find the value of the polynomial \(5x - 4x^2 + 3\) at:
(i) \(x = 0\)
(ii) \(x = -1\)
(iii) \(x = 2\)
(i) 3
(ii) -6
(iii) -3
Find \(p(0)\), \(p(1)\), and \(p(2)\) for each of the following polynomials:
(i) \(p(y) = y^2 - y + 1\)
(ii) \(p(t) = 2 + t + 2t^2 - t^3\)
(iii) \(p(x) = x^3\)
(iv) \(p(x) = (x - 1)(x + 1)\)
(i) 1, 1, 3
(ii) 2, 4, 4
(iii) 0, 1, 8
(iv) -1, 0, 3
Verify whether the following are zeroes of the polynomial, as indicated:
(i) \(p(x) = 3x + 1\), \(x = -\dfrac{1}{3}\)
(ii) \(p(x) = 5x - \pi\), \(x = \dfrac{4}{5}\)
(iii) \(p(x) = x^2 - 1\), \(x = 1, -1\)
(iv) \(p(x) = (x + 1)(x - 2)\), \(x = -1, 2\)
(v) \(p(x) = x^2\), \(x = 0\)
(vi) \(p(x) = lx + m\), \(x = -\dfrac{m}{l}\)
(vii) \(p(x) = 3x^2 - 1\), \(x = -\dfrac{1}{\sqrt{3}}, \dfrac{2}{\sqrt{3}}\)
(viii) \(p(x) = 2x + 1\), \(x = \dfrac{1}{2}\)
(i) Yes
(ii) No
(iii) Yes
(iv) Yes
(v) Yes
(vi) Yes
(vii) \(-\dfrac{1}{\sqrt{3}}\) is a zero, but \(\dfrac{2}{\sqrt{3}}\) is not a zero
(viii) No
Find the zero of the polynomial in each of the following cases:
(i) \(p(x) = x + 5\)
(ii) \(p(x) = x - 5\)
(iii) \(p(x) = 2x + 5\)
(iv) \(p(x) = 3x - 2\)
(v) \(p(x) = 3x\)
(vi) \(p(x) = ax\), \(a \neq 0\)
(vii) \(p(x) = cx + d\), where \(c, d\) are real numbers
(i) -5
(ii) 5
(iii) -\dfrac{5}{2}
(iv) \dfrac{2}{3}
(v) 0
(vi) 0
(vii) -\dfrac{d}{c}
Determine which of the following polynomials has \((x + 1)\) as a factor:
(i) \(x^3 + x^2 + x + 1\)
(ii) \(x^4 + x^3 + x^2 + x + 1\)
(iii) \(x^4 + 3x^3 + 3x^2 + x + 1\)
(iv) \(x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}\)
(x + 1) is a factor of (i), but not a factor of (ii), (iii), and (iv).
Use the Factor Theorem to determine whether \(g(x)\) is a factor of \(p(x)\) in each case:
(i) \(p(x) = 2x^3 + x^2 - 2x - 1\), \(g(x) = x + 1\)
(ii) \(p(x) = x^3 + 3x^2 + 3x + 1\), \(g(x) = x + 2\)
(iii) \(p(x) = x^3 - 4x^2 + x + 6\), \(g(x) = x - 3\)
(i) Yes
(ii) No
(iii) Yes
Find the value of \(k\), if \(x - 1\) is a factor of \(p(x)\) in each of the following cases:
(i) \(p(x) = x^2 + x + k\)
(ii) \(p(x) = 2x^2 + kx + \sqrt{2}\)
(iii) \(p(x) = kx^2 - \sqrt{2}x + 1\)
(iv) \(p(x) = kx^2 - 3x + k\)
(i) -2
(ii) \(-(2 + \sqrt{2})\)
(iii) \(\sqrt{2} - 1\)
(iv) \(\dfrac{3}{2}\)
Factorise the following:
(i) \(12x^2 - 7x + 1\)
(ii) \(2x^2 + 7x + 3\)
(iii) \(6x^2 + 5x - 6\)
(iv) \(3x^2 - x - 4\)
(i) \((3x - 1)(4x - 1)\)
(ii) \((x + 3)(2x + 1)\)
(iii) \((2x + 3)(3x - 2)\)
(iv) \((x + 1)(3x - 4)\)
Factorise the following:
(i) \(x^3 - 2x^2 - x + 2\)
(ii) \(x^3 - 3x^2 - 9x - 5\)
(iii) \(x^3 + 13x^2 + 32x + 20\)
(iv) \(2y^3 + y^2 - 2y - 1\)
(i) \((x - 2)(x - 1)(x + 1)\)
(ii) \((x + 1)(x + 1)(x - 5)\)
(iii) \((x + 1)(x + 2)(x + 10)\)
(iv) \((y - 1)(y + 1)(2y + 1)\)
Use suitable identities to find the following products:
(i) \((x + 4)(x + 10)\)
(ii) \((x + 8)(x - 10)\)
(iii) \((3x + 4)(3x - 5)\)
(iv) \((y^2 + \dfrac{3}{2})(y^2 - \dfrac{3}{2})\)
(v) \((3 - 2x)(3 + 2x)\)
(i) \(x^2 + 14x + 40\)
(ii) \(x^2 - 2x - 80\)
(iii) \(9x^2 - 3x - 20\)
(iv) \(y^4 - \dfrac{9}{4}\)
(v) 9 - 4x^2
Evaluate the following products without multiplying directly:
(i) 103 × 107
(ii) 95 × 96
(iii) 104 × 96
(i) 11021
(ii) 9120
(iii) 9984
Factorise the following using appropriate identities:
(i) \(9x^2 + 6xy + y^2\)
(ii) \(4y^2 - 4y + 1\)
(iii) \(x^2 - \dfrac{y^2}{100}\)
(i) \((3x + y)(3x + y)\)
(ii) \((2y - 1)(2y - 1)\)
(iii) \((x + \dfrac{y}{10})(x - \dfrac{y}{10})\)
Expand each of the following using suitable identities:
(i) \((x + 2y + 4z)^2\)
(ii) \((2x - y + z)^2\)
(iii) \((-2x + 3y + 2z)^2\)
(iv) \((3a - 7b - c)^2\)
(v) \((-2x + 5y - 3z)^2\)
(vi) \([\dfrac{1}{4}a - \dfrac{1}{2}b + 1]^2\)
(i) \(x^2 + 4y^2 + 16z^2 + 4xy + 16yz + 8xz\)
(ii) \(4x^2 + y^2 + z^2 - 4xy - 2yz + 4xz\)
(iii) \(4x^2 + 9y^2 + 4z^2 - 12xy + 12yz - 8xz\)
(iv) \(9a^2 + 49b^2 + c^2 - 42ab + 14bc - 6ac\)
(v) \(4x^2 + 25y^2 + 9z^2 - 20xy - 30yz + 12xz\)
(vi) \(\dfrac{a^2}{16} + \dfrac{b^2}{4} + 1 - \dfrac{ab}{4} - b + \dfrac{a}{2}\)
Factorise:
(i) \((2x + 3y - 4z)(2x + 3y - 4z)\)
(ii) \(( -\sqrt{2}x + y + 2\sqrt{2}z)( -\sqrt{2}x + y + 2\sqrt{2}z)\)
(i) \(8x^3 + 12x^2y + 6x + 1\)
(ii) \(8a^3 - 27b^3 - 36a^2b + 54ab^2\)
Write the following cubes in expanded form:
(i) \((2x + 1)^3\)
(ii) \((2a - 3b)^3\)
(iii) \([ \dfrac{3}{2}x + 1]^3\)
(iv) \([x - \dfrac{2}{3}y]^3\)
(i) \(8x^3 + 27/8 x^3 + 27/4 x^2 + 9/2 x + 1\)
(ii) \(x^3 - \dfrac{8}{27}y^3 - 2x^2y + \dfrac{4xy^2}{3}\)
Evaluate the following using suitable identities:
(i) \((99)^3\)
(ii) \((102)^3\)
(iii) \((998)^3\)
(i) 970299
(ii) 1061208
(iii) 994011992
Factorise each of the following:
(i) \(8a^3 + b^3 + 12a^2b + 6ab^2\)
(ii) \(8a^3 - b^3 - 12a^2b + 6ab^2\)
(iii) \(27 - 125a^3 - 135a + 225a^2\)
(iv) \(64a^3 - 27b^3 - 144a^2b + 108ab^2\)
(v) \(27p^3 - \dfrac{1}{216} - \dfrac{9}{2}p^2 + \dfrac{1}{4}p\)
(i) (2a + b)(2a + b)(2a + b)
(ii) (2a - b)(2a - b)(2a - b)
(iii) (3 - 5a)(3 - 5a)(3 - 5a)
(iv) (4a - 3b)(4a - 3b)(4a - 3b)
(v) (3p - 1/6)(3p - 1/6)(3p - 1/6)
Verify:
(i) \(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\)
(ii) \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\)
Simplify RHS.
Factorise the following:
(i) \(27y^3 + 125z^3\)
(ii) \(64m^3 - 343n^3\)
(i) \((3y + 5z)(9y^2 - 15yz + 25z^2)\)
(ii) \((4m - 7n)(16m^2 + 49n^2 + 28mn)\)
Factorise: \(27x^3 + y^3 + z^3 - 9xyz\)
(3x + y + z)(9x^2 + y^2 + z^2 - 3xy - yz - 3xz)
Verify that \(x^3 + y^3 + z^3 - 3xyz = \dfrac{1}{2}(x + y + z)[(x - y)^2 + (y - z)^2 + (z - x)^2]\)
Simplify RHS.
If \(x + y + z = 0\), show that \(x^3 + y^3 + z^3 = 3xyz\)
Put \(x + y + z = 0\) in the identity in Q12.
Without calculating cubes, find the value of each:
(i) \((-12)^3 + (7)^3 + (5)^3\)
(ii) \((28)^3 + (-15)^3 + (-13)^3\)
(i) -1260
(ii) 16380
Give possible expressions for the length and breadth of rectangles whose areas are:
(i) \(25a^2 - 35a + 12\)
(ii) \(35y^2 + 13y - 12\)
(i) One possible answer: Length = \(5a - 3\), Breadth = \(5a - 4\)
(ii) One possible answer: Length = \(7y - 3\), Breadth = \(5y + 4\)
Find possible expressions for the dimensions of cuboids whose volumes are:
(i) \(3x^2 - 12x\)
(ii) \(12ky^2 + 8ky - 20k\)
(i) One possible answer: \(3x, x, x - 4\)
(ii) One possible answer: \(4k, 3y + 5, y - 1\)